Abstract
Guided by the symmetries of the Euler–Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation—called second-order, Euler–Lagrange vector fields (SOELVFs)—with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.
Highlights
The Lagrangian phase space formulation of mechanics [1–4], with its roots in differential geometry, provides an especially fruitful framework with which to analyze dynamical systems of singular Lagrangians L
We begin with Lagrangian mechanics, and an analysis of the generalized Lie symmetry
The first-order constraints, those that come directly from the energy equation, are the focus of this section. Most of this analysis is done for a SOLVF, we show later that our results do not depend on this choice
Summary
The Lagrangian phase space formulation of mechanics [1–4], with its roots in differential geometry, provides an especially fruitful framework with which to analyze dynamical systems of singular Lagrangians L. It is known that the Lagrangian constraints obtained while a constraint algorithm is being imposed on a SOLVF need not be projectable [4, 11, 16, 18, 19, 22, 29] Combined, this means that dynamics on the Lagrangian phase space and dynamics on the Hamiltonian phase space can take place on two inequivalent submanifolds, be determined by two inequivalent vector fields, resulting in two different families of trajectories on the configuration space Q for the same dynamical system with the same initial data. We are led to construct the second-order, Euler-Lagrange vector field (SOELVF) These fields avoid the second-order problem, are projectable to the Hamiltonian phase space, and lie on Lagrangian constraint submanifolds that are projectable.
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